Convergence of the Allen-Cahn equation with a zero Neumann boundary condition on non-convex domains

TL;DR

This paper proves that solutions to the Allen-Cahn equation with Neumann boundary conditions on non-convex domains converge to Brakke's mean curvature flow, extending understanding of phase transitions in complex geometries.

Contribution

It establishes the convergence of the Allen-Cahn equation to mean curvature flow with boundary conditions on non-convex domains, a novel extension of existing results.

Findings

Convergence of diffused surface energy to Brakke's flow.

Validation of generalized right angle boundary condition.

Applicability to non-convex domains with smooth boundaries.

Abstract

We study a singular limit problem of the Allen-Cahn equation with the homogeneous Neumann boundary condition on non-convex domains with smooth boundaries under suitable assumptions for initial data. The main result is the convergence of the time parametrized family of the diffused surface energy to Brakke's mean curvature flow with a generalized right angle condition on the boundary of the domain.